Question: Solve for $t$, $ \dfrac{1}{5t - 20} = \dfrac{2t - 2}{t - 4} + \dfrac{3}{3t - 12} $
Solution: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5t - 20$ $t - 4$ and $3t - 12$ The common denominator is $15t - 60$ To get $15t - 60$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{1}{5t - 20} \times \dfrac{3}{3} = \dfrac{3}{15t - 60} $ To get $15t - 60$ in the denominator of the second term, multiply it by $\frac{15}{15}$ $ \dfrac{2t - 2}{t - 4} \times \dfrac{15}{15} = \dfrac{30t - 30}{15t - 60} $ To get $15t - 60$ in the denominator of the third term, multiply it by $\frac{5}{5}$ $ \dfrac{3}{3t - 12} \times \dfrac{5}{5} = \dfrac{15}{15t - 60} $ This give us: $ \dfrac{3}{15t - 60} = \dfrac{30t - 30}{15t - 60} + \dfrac{15}{15t - 60} $ If we multiply both sides of the equation by $15t - 60$ , we get: $ 3 = 30t - 30 + 15$ $ 3 = 30t - 15$ $ 18 = 30t $ $ t = \dfrac{3}{5}$